## 44001

44001^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 44016

44016^{2} = 1937408256, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 44032

44032^{2} = (1^{2} + 7)(5^{2} + 7)(6^{2} + 7)(11^{2} + 7)(37^{2} + 7) = (6^{2} + 7)(37^{2} + 7)(181^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 44047

44047^{2} = 1536^{2} + 1537^{2} + 1538^{2} + 1539^{2} + ... + 2113^{2}.

by Yoshio Mimura, Kobe, Japan

## 44048

1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + 44048^{2} = 28488666200024, which consists of even digits.

by Yoshio Mimura, Kobe, Japan

## 44049

44049^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 44064

435^{2} + 44064 = 483^{2}, 435^{2} - 44064 = 381^{2}.

44064^{2} = (1^{2} + 8)(2^{2} + 8)(3^{2} + 8)(4^{2} + 8)(10^{2} + 8)(20^{2} + 8)

= (1^{2} + 8)(2^{2} + 8)(8^{2} + 8)(10^{2} + 8)(48^{2} + 8) = (1^{2} + 8)(4^{2} + 8)(10^{2} + 8)(14^{2} + 8)(20^{2} + 8)

= (2^{2} + 8)(3^{2} + 8)(4^{2} + 8)(14^{2} + 8)(44^{2} + 8) = (2^{2} + 8)(14^{2} + 8)(20^{2} + 8)(44^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 44065

44065^{2} = (13^{2} + 6)(3331^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 44070

44070^{2} = (2^{2} + 9)(45^{2} + 9)(271^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 44079

The quadratic polynomial -44079X^{2} + 416346X - 173351 takes the values 446^{2}, 695^{2}, 824^{2}, 887^{2}, 898^{2}, 859^{2} at X = 1, 2,..., 6

by Yoshio Mimura, Kobe, Japan

## 44110

44110^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 44118

44118^{2} = 3191 x 3192 + 3193 x 3194 + 3195 x 3196 + 3197 x 3198 + ... + 3533 x 3534.

44118^{2} = (1^{2} + 2)(6^{2} + 2)(59^{2} + 2)(70^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 44127

1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + 44127^{2} = 28642222628880, which consists of even digits.

by Yoshio Mimura, Kobe, Japan

## 44138

44138^{2} = (1^{2} + 1)(39^{2} + 1)(800^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 44200

44200^{2} = (1^{2} + 1)(3^{2} + 1)(5^{2} + 1)(7^{2} + 1)(13^{2} + 1)(21^{2} + 1)

= (1^{2} + 4)(2^{2} + 4)(9^{2} + 4)(26^{2} + 4)(29^{2} + 4) = (1^{2} + 4)(3^{2} + 4)(4^{2} + 4)(8^{2} + 4)(9^{2} + 4)(16^{2} + 4)

= (1^{2} + 4)(3^{2} + 4)(4^{2} + 4)(16^{2} + 4)(76^{2} + 4) = (1^{2} + 1)(3^{2} + 1)(5^{2} + 1)(7^{2} + 1)(13^{2} + 1)(21^{2} + 1)

= (1^{2} + 4)(3^{2} + 4)(6^{2} + 4)(9^{2} + 4)(94^{2} + 4) = (1^{2} + 4)(3^{2} + 4)(9^{2} + 4)(14^{2} + 4)(42^{2} + 4)

= (1^{2} + 4)(4^{2} + 4)(8^{2} + 4)(536^{2} + 4) = (2^{2} + 4)(3^{2} + 4)(9^{2} + 4)(11^{2} + 4)(42^{2} + 4)

= (3^{2} + 4)(4^{2} + 4)(8^{2} + 4)(9^{2} + 4)(36^{2} + 4) = (3^{2} + 4)(4^{2} + 4)(36^{2} + 4)(76^{2} + 4)

= (3^{2} + 4)(9^{2} + 4)(14^{2} + 4)(94^{2} + 4) = (3^{2} + 4)(11^{2} + 4)(26^{2} + 4)(42^{2} + 4)

= (6^{2} + 4)(9^{2} + 4)(26^{2} + 4)(29^{2} + 4) = (8^{2} + 4)(9^{2} + 4)(16^{2} + 4)(36^{2} + 4)

= (9^{2} + 4)(10^{2} + 4)(11^{2} + 4)(42^{2} + 4) = (9^{2} + 4)(42^{2} + 4)(114^{2} + 4)

= (11^{2} + 4)(42^{2} + 4)(94^{2} + 4) = (16^{2} + 4)(36^{2} + 4)(76^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 44244

44244^{2} = 1957531536, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 44254

44254^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 44275

44275^{2} = 1960275625, and 196 = 14^{2}, 275625 = 525^{2}.

by Yoshio Mimura, Kobe, Japan

## 44319

44319^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 44352

44352^{2} = (7^{2} - 1)(10^{2} - 1)(15^{2} - 1)(43^{2} - 1) = (10^{2} - 1)(13^{2} - 1)(15^{2} - 1)(23^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 44367

44367^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 44370

44370^{2} = (1^{2} + 9)(6^{2} + 9)(7^{2} + 9)(12^{2} + 9)(22^{2} + 9) = (1^{2} + 9)(12^{2} + 9)(22^{2} + 9)(51^{2} + 9)

= (1^{2} + 9)(114^{2} + 9)(123^{2} + 9) = (3^{2} + 9)(4^{2} + 9)(7^{2} + 9)(12^{2} + 9)(22^{2} + 9)

= (3^{2} + 9)(6^{2} + 9)(1559^{2} + 9) = (3^{2} + 9)(22^{2} + 9)(471^{2} + 9) = (4^{2} + 9)(5^{2} + 9)(12^{2} + 9)(123^{2} + 9)

= (5^{2} + 9)(6^{2} + 9)(22^{2} + 9)(51^{2} + 9) = (5^{2} + 9)(9^{2} + 9)(22^{2} + 9)(36^{2} + 9)

= (6^{2} + 9)(7^{2} + 9)(22^{2} + 9)(39^{2} + 9) = (7^{2} + 9)(12^{2} + 9)(21^{2} + 9)(22^{2} + 9)

= (7^{2} + 9)(12^{2} + 9)(471^{2} + 9) = (7^{2} + 9)(51^{2} + 9)(114^{2} + 9) = (12^{2} + 9)(29^{2} + 9)(123^{2} + 9)

= (22^{2} + 9)(39^{2} + 9)(51^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 44376

215^{2} + 44376 = 301^{2}, 215^{2} - 44376 = 43^{2}.

by Yoshio Mimura, Kobe, Japan

## 44457

44457^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 44460

44460^{2} = (2^{2} - 1)(4^{2} - 1)(39^{2} - 1)(170^{2} - 1)
44460,(2^{2} - 1)(151^{2} - 1)(170^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 44522

44522^{2} = (14^{2} + 1)(15^{2} + 1)(211^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 44523

44523^{2} = (1^{2} + 2)(7^{2} + 2)(17^{2} + 2)(211^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 44528

44528^{2} = (2^{2} + 7)(4^{2} + 7)(13^{2} + 7)(211^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 44590

44590^{2} = 995^{2} + 997^{2} + 999^{2} + 1001^{2} + ... + 2345^{2}.

by Yoshio Mimura, Kobe, Japan

## 44623

44623^{2} = 1991212129, a square with 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 44632

44632^{2} = 716 x 717 + 718 x 719 + 720 x 721 + 722 x 723 + 724 x 725+ ... + 2308 x 2309.

by Yoshio Mimura, Kobe, Japan

## 44642

44642^{2} = (1^{2} + 1)(10^{2} + 1)(3141^{2} + 1)

= (1^{2} + 1)(5^{2} + 1)(10^{2} + 1)(616^{2} + 1) = (4^{2} + 1)(21^{2} + 1)(515^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 44654

44654^{2} = 1993979716, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 4655

44657^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 44660

44660^{2} = (1^{2} + 6)(2^{2} + 6)(4^{2} + 6)(8^{2} + 6)(136^{2} + 6) = (2^{2} + 6)(4^{2} + 6)(22^{2} + 6)(136^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 44678

44678^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 44688

44688^{2} = (1^{2} + 3)(2^{2} + 3)(3^{2} + 3)(4^{2} + 3)(9^{2} + 3)(61^{2} + 3)

= (1^{2} + 3)(2^{2} + 3)(3^{2} + 3)(4^{2} + 3)(15^{2} + 3)(37^{2} + 3) = (1^{2} + 3)(2^{2} + 3)(9^{2} + 3)(15^{2} + 3)(61^{2} + 3)

= (1^{2} + 3)(3^{2} + 3)(4^{2} + 3)(5^{2} + 3)(12^{2} + 3)(23^{2} + 3) = (1^{2} + 3)(4^{2} + 3)(9^{2} + 3)(15^{2} + 3)(37^{2} + 3)

= (1^{2} + 3)(5^{2} + 3)(12^{2} + 3)(15^{2} + 3)(23^{2} + 3) = (2^{2} + 3)(3^{2} + 3)(4^{2} + 3)(5^{2} + 3)(9^{2} + 3)(23^{2} + 3)

= (2^{2} + 3)(3^{2} + 3)(5^{2} + 3)(15^{2} + 3)(61^{2} + 3) = (2^{2} + 3)(5^{2} + 3)(9^{2} + 3)(15^{2} + 3)(23^{2} + 3)

= (3^{2} + 3)(4^{2} + 3)(5^{2} + 3)(9^{2} + 3)(61^{2} + 3) = (3^{2} + 3)(4^{2} + 3)(5^{2} + 3)(15^{2} + 3)(37^{2} + 3)

= (3^{2} + 3)(9^{2} + 3)(23^{2} + 3)(61^{2} + 3) = (3^{2} + 3)(15^{2} + 3)(23^{2} + 3)(37^{2} + 3)

= (5^{2} + 3)(9^{2} + 3)(15^{2} + 3)(61^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 44704

44704^{2} = (2^{2} + 7)(45^{2} + 7)(299^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 44734

(44734^{2} +2) = (1^{2} + 2)(3^{2} + 2)(4^{2} + 2)(7^{2} + 2)(15^{2} + 2)(17^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 44798

44798^{2} = 2006860804, a square with even digits.

by Yoshio Mimura, Kobe, Japan

## 44814

44814^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 44884

44884^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 44913

44913^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 44928

44928^{2} = (2^{2} - 1)(3^{2} - 1)(7^{2} - 1)(25^{2} - 1)(53^{2} - 1) = (5^{2} - 1)(7^{2} - 1)(25^{2} - 1)(53^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 44936

44936^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 44946

44946^{2} = (1^{2} + 2)(2^{2} + 2)(3^{2} + 2)(15^{2} + 2)(212^{2} + 2) = (1^{2} + 2)(8^{2} + 2)(15^{2} + 2)(212^{2} + 2)

= (3^{2} + 2)(4^{2} + 2)(15^{2} + 2)(212^{2} + 2) = (14^{2} + 2)(15^{2} + 2)(212^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 44950

44950^{2} = (2^{2} + 6)(5^{2} + 6)(38^{2} + 6)(67^{2} + 6) = (2^{2} + 6)(67^{2} + 6)(212^{2} + 6)

= (5^{2} + 6)(38^{2} + 6)(212^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 44954

44954^{2} = (2^{2} + 3)(124^{2} + 3)(137^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 44978

44978^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 44944

44944 = 212^{2}, a palindromic square with 2 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 44988

44988^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 44990

44990^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 44992

44992^{2} = 2024280064, a square with even digits.

by Yoshio Mimura, Kobe, Japan

## 44998

44998^{2} = 2024820004, a square with even digits.

by Yoshio Mimura, Kobe, Japan